This documentation is automatically generated by online-judge-tools/verification-helper
#include "graph/others/low-link.hpp"
橋や関節点などを効率的に求める際に有効なアルゴリズム.
グラフをDFSして各頂点 idx
について, ord[idx]
:= DFS で頂点に訪れた順番, low[idx]
:= 頂点 $idx$ からDFS木の葉方向の辺を $0$ 回以上, 後退辺を $1$ 回以下通って到達可能な頂点の ord
の最小値 を求める.
ある頂点 $u$ が関節点であるとき, DFS木の根については子が $2$ つ以上, それ以外の頂点については 頂点 $u$ のある子 $v$ について ord[u]
$\le$ low[v]
を満たす.
ある辺 $(u, v)$ が橋であるとき, ord[u]
$\lt$ low[v]
を満たす.
build()
: LowLink を構築する. 構築後, articulation
には関節点, bridge
には橋が格納される. 非連結でも多重辺を含んでいてもOK.$O(V + E)$
#pragma once
#include "../graph-template.hpp"
/**
* @brief Low Link(橋/関節点)
* @see http://kagamiz.hatenablog.com/entry/2013/10/05/005213
*
*/
template <typename T = int>
struct LowLink : Graph<T> {
public:
using Graph<T>::Graph;
vector<int> ord, low, articulation;
vector<Edge<T> > bridge;
using Graph<T>::g;
virtual void build() {
used.assign(g.size(), 0);
ord.assign(g.size(), 0);
low.assign(g.size(), 0);
int k = 0;
for (int i = 0; i < (int)g.size(); i++) {
if (!used[i]) k = dfs(i, k, -1);
}
}
explicit LowLink(const Graph<T> &g) : Graph<T>(g) {}
private:
vector<int> used;
int dfs(int idx, int k, int par) {
used[idx] = true;
ord[idx] = k++;
low[idx] = ord[idx];
bool is_articulation = false, beet = false;
int cnt = 0;
for (auto &to : g[idx]) {
if (to == par && !exchange(beet, true)) {
continue;
}
if (!used[to]) {
++cnt;
k = dfs(to, k, idx);
low[idx] = min(low[idx], low[to]);
is_articulation |= par >= 0 && low[to] >= ord[idx];
if (ord[idx] < low[to]) bridge.emplace_back(to);
} else {
low[idx] = min(low[idx], ord[to]);
}
}
is_articulation |= par == -1 && cnt > 1;
if (is_articulation) articulation.push_back(idx);
return k;
}
};
#line 2 "graph/others/low-link.hpp"
#line 2 "graph/graph-template.hpp"
/**
* @brief Graph Template(グラフテンプレート)
*/
template <typename T = int>
struct Edge {
int from, to;
T cost;
int idx;
Edge() = default;
Edge(int from, int to, T cost = 1, int idx = -1)
: from(from), to(to), cost(cost), idx(idx) {}
operator int() const { return to; }
};
template <typename T = int>
struct Graph {
vector<vector<Edge<T> > > g;
int es;
Graph() = default;
explicit Graph(int n) : g(n), es(0) {}
size_t size() const { return g.size(); }
void add_directed_edge(int from, int to, T cost = 1) {
g[from].emplace_back(from, to, cost, es++);
}
void add_edge(int from, int to, T cost = 1) {
g[from].emplace_back(from, to, cost, es);
g[to].emplace_back(to, from, cost, es++);
}
void read(int M, int padding = -1, bool weighted = false,
bool directed = false) {
for (int i = 0; i < M; i++) {
int a, b;
cin >> a >> b;
a += padding;
b += padding;
T c = T(1);
if (weighted) cin >> c;
if (directed)
add_directed_edge(a, b, c);
else
add_edge(a, b, c);
}
}
inline vector<Edge<T> > &operator[](const int &k) { return g[k]; }
inline const vector<Edge<T> > &operator[](const int &k) const { return g[k]; }
};
template <typename T = int>
using Edges = vector<Edge<T> >;
#line 4 "graph/others/low-link.hpp"
/**
* @brief Low Link(橋/関節点)
* @see http://kagamiz.hatenablog.com/entry/2013/10/05/005213
*
*/
template <typename T = int>
struct LowLink : Graph<T> {
public:
using Graph<T>::Graph;
vector<int> ord, low, articulation;
vector<Edge<T> > bridge;
using Graph<T>::g;
virtual void build() {
used.assign(g.size(), 0);
ord.assign(g.size(), 0);
low.assign(g.size(), 0);
int k = 0;
for (int i = 0; i < (int)g.size(); i++) {
if (!used[i]) k = dfs(i, k, -1);
}
}
explicit LowLink(const Graph<T> &g) : Graph<T>(g) {}
private:
vector<int> used;
int dfs(int idx, int k, int par) {
used[idx] = true;
ord[idx] = k++;
low[idx] = ord[idx];
bool is_articulation = false, beet = false;
int cnt = 0;
for (auto &to : g[idx]) {
if (to == par && !exchange(beet, true)) {
continue;
}
if (!used[to]) {
++cnt;
k = dfs(to, k, idx);
low[idx] = min(low[idx], low[to]);
is_articulation |= par >= 0 && low[to] >= ord[idx];
if (ord[idx] < low[to]) bridge.emplace_back(to);
} else {
low[idx] = min(low[idx], ord[to]);
}
}
is_articulation |= par == -1 && cnt > 1;
if (is_articulation) articulation.push_back(idx);
return k;
}
};